Shortcut to synchronization in classical and quantum systems

Synchronization is a major concept in nonlinear physics. In a large number of systems, it is observed at long times for a sinusoidal excitation. In this paper, we design a transiently non-sinusoidal driving to reach the synchronization regime more quickly. We exemplify an inverse engineering method to solve this issue on the classical Van der Pol oscillator. This approach cannot be directly transposed to the quantum case as the system is no longer point-like in phase space. We explain how to adapt our method by an iterative procedure to account for the finite-size quantum distribution in phase space. We show that the resulting driving yields a density matrix close to the synchronized one according to the trace distance. Our method provides an example of fast control of a nonlinear quantum system, and raises the question of the quantum speed limit concept in the presence of nonlinearities.


Shortcut-to-synchronization in a classical van der Pol oscillator
We detail here the procedure to design of a shortcut-to-synchronization from the initial point (x 0 , y 0 ) = (0, 0). We first solve the trajectory of a sinusoidally driven van-der-Pol oscillator with ε 0 = 1.5/T 0 and obtain the branching point (x ∞ , y ∞ ) ≃ (0.29, 1.05) corresponding to t ∞ = 50.125 × T 0 . The proximity of this branching point to the "vertical" of the initial position (x 0 , y 0 ) enables a fast shortcut with the amplitude ε(t).
We first define a system trajectory of the form y short, The chosen trajectory fulfills, for any value of the parameter γ, the required boundary conditions y short,γ (0) = y 0 and y short,γ (τ) = y ∞ associated respectively to the initial and final shortcut times. The additional condition y ′ short,γ (τ) = y ′ 0 (t ∞ ) provides a continuity of the driving amplitude between the transient and sinusoidal part. To determine the correct shortcut trajectory and fix the γ parameter, we use a self-consistency argument: by virtue of Eq.(1) of the main text, when the system goes along the trajectory y short,γ (t), the coordinate motion x(t) follows a differential equation where y short,γ (t) acts as a driving term: With the considered initial condition x(0) = x 0 = 0, each value of γ yields a corresponding solution x short,γ (t) and final coordinate x short,γ (τ) at the time τ. For the "magic"value γ 0 , the final coordinate reaches the target, i.e. x short,γ 0 (τ) = x ∞ . Then, the trajectory (x short,γ 0 (t), y short,γ 0 (t)) reaches the branching point (x ∞ , y ∞ ) at t = τ, and can thus be choses as shortcut trajectory. For the parameters above, one finds numerically γ 0 ≃ −9.3532. The corresponding driving amplitude ε short (t) is derived from Eq.
(1) of the main text as for t ≤ τ. For t > τ, the sinusoidal driving is resumed ε(t) = ε 0 cos(ωt + ϕ). The phase ϕ is fixed as follows. At time τ, the system is at a position that would be reached under a plain sinusoidal driving ε 0 (t) at time t ∞ . For our strategy to be sucessful, the system must be subject to a driving ε(t) such that

Shortcut-to-synchronization in a quantum van der Pol oscillator
We detail the procedure for the shortcuts considered in Figs. 2 and 3 in the weakly/strongly non-linear regimes. We solve Eq.
(2) of the main text in a quantum subspace corresponding to the N lowest-energy level of the harmonic oscillator. It is sufficient to consider N = 40, as higher-energy quantum states are irrelevant for the considered initial states and Hamiltonians.